Sums of squares II: Matrix functions

Matrix Completion , Free Resolutions , and Sums of Squares

Goal: Describe the image of the cone Sn ≥� of positive semidefinite quadratic forms under the projection πG . Sn ≥�: convex cone of quadratic forms∑i , j ai jxix j such that∑i , j ai jpi p j ≥ � for all (p�, . . . , pn) ∈ Rn. Theorem (Diagonalization ofQuadratic Forms). Aquadratic form q ∈ R[x�, . . . , xn] is positive semidefinite if and only if it is a sum of squares of linear forms after a c...

Uncorrected Sums of Squares and Cross Products Matrix (USSCP)

The Data Matrix The most important matrix for any statistical procedure is the data matrix. The observations form the rows of the data matrix and the variables form the columns. The most important requirement for the data matrix is that the rows of the matrix should be statistically independent. That is, if we pick any single row of the data matrix, then we should not be able to predict any oth...

Pure States, Positive Matrix Polynomials and Sums of Hermitian Squares

Let M be an archimedean quadratic module of real t× t matrix polynomials in n variables, and let S ⊆ R be the set of all points where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and S × P(R). This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol a...

Sums of Squares of Regular Functions on Real Algebraic Varieties

Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dimV ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same h...

Sums of distinct squares